Theorem imaeq2 5064

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 5000	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4953	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4732	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4732	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1395   ran crn 4720   |` cres 4721   "cima 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  imaeq2i  5066  imaeq2d  5068  fimadmfo  5559  ssimaex  5697  ssimaexg  5698  isoselem  5950  f1opw2  6218  fopwdom  7005  ssenen  7020  fiintim  7104  fidcenumlemrk  7132  fidcenumlemr  7133  sbthlem2  7136  isbth  7145  ennnfonelemp1  12993  ennnfonelemnn0  13009  ctinfomlemom  13014  ctinfom  13015  tgcn  14898  iscnp4  14908  cnpnei  14909  cnima  14910  cnconst2  14923  cnrest2  14926  cnptoprest  14929  txcnp  14961  txcnmpt  14963  metcnp3  15201